   Chapter 11.R, Problem 19E

Chapter
Section
Textbook Problem

Determine whether the series is convergent or divergent. ∑ n = 1 ∞ 1 ⋅ 3 ⋅ 5 ⋅ ⋅ ⋅ ⋅ ⋅ ( 2 n − 1 ) 5 n n !

To determine

Whether the given series is convergent or divergent

Explanation

1) Concept:

The Ratio Test

i) If  limnan+1an=L<1, then the series n=1an is absolutely convergent (and therefore convergent).

ii) If  limnan+1an=L>1 or  n=1an+1an=, then the series n=1an is divergent.

iii) If  limnan+1an=1, the Ratio Test is inconclusive.

If a series an  is absolutely convergent, then it is convergent.

2) Given:

n=11·3·5··2n-15nn!

3) Calculation:

It is given that

n=11·3·5··2n-15nn!

Consider an=1·3·5··2n-15nn!.

To write  an+1:

an+1=1·3·5··2n+1-15n+1n+1!

=1·3·5··2n+15n+1n+1!

limn

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